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Tag 02AW

Chapter 102: Exercises > Section 102.49: Schemes, Final Exam, Spring 2009

Exercise 102.49.1. Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point. Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.

  1. Show that $x$ is a closed point of $X$.
  2. Show that $H^0(X, \mathcal{F})$ is not zero.
  3. Show that $\mathcal{F}$ is generated by global sections.
  4. Show that $H^p(X, \mathcal{F}) = 0$ for $p > 0$.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 4925–4937 (see updates for more information).

    \begin{exercise}
    \label{exercise-Noetherian-coherent}
    Let $X$ be a Noetherian scheme.
    Let $\mathcal{F}$ be a coherent sheaf on $X$.
    Let $x \in X$ be a point.
    Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.
    \begin{enumerate}
    \item Show that $x$ is a closed point of $X$.
    \item Show that $H^0(X, \mathcal{F})$ is not zero.
    \item Show that $\mathcal{F}$ is generated by global sections.
    \item Show that $H^p(X, \mathcal{F}) = 0$ for $p > 0$.
    \end{enumerate}
    \end{exercise}

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